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First moments of ${\rm{GL}} (3) \times {\rm{GL}} (2)$ and ${\rm{GL}} (2)$ $L$-functions and their applications

Fei Hou

TL;DR

This work advances the subconvexity problem for high-riber L-functions by establishing first-moment bounds for GL(3)×GL(2) and GL(2) L-functions in both level and weight aspects. It combines amplification, Petersson trace theory, and GL(3) Voronoï summation with reciprocity to control intricate bilinear forms and Kloosterman sums, carefully addressing root-number contributions. The main contributions include a level-aspect subconvex bound for L(1/2, F\otimes f) in a hybrid prime-level range, a Lindelöf-mean bound for the weight- aspect first moment of L-factors, and a power-saving asymptotic formula in the weight variable K, with non-vanishing conclusions as a by-product. The results provide new tools for studying equidistribution of L^q-mass, Heegner points, and QUE for higher-rank automorphic L-functions, and they illustrate a path toward simultaneous subconvexity in multiple automorphic parameters.

Abstract

Let $F$ be a self-dual Hecke-Maaß form for ${\rm{GL}}(3)$ underlying the symmetric square lift of a ${\rm{GL}}(2)$-newform of square-free level and trivial nebentypus. In this paper, we are interested in the first moments of the central values of ${\rm{GL}}(3) \times {\rm{GL}}(2)$ $L$-functions and ${\rm{GL}}(2)$ $L$-functions. As a result, we obtain an estimate for the first moment for $L(1/2, F\otimes f)$ over a family, where $F$ is of the level $q^2$, and $f\in \mathcal{B}^\ast_k(M)$ for any primes $q,M\ge 2$ such that $(q,M)=1$. We prove the subconvex bound for $L(1/2, F\otimes f)$ involving the levels aspects simultaneously in the range $M^{13/64+\varepsilon }\le q \le M^{11/40-\varepsilon}$ and $M> q^δ$ for any $\varepsilon, δ>0$ for the first time. Moreover, we further investigate the first moments of these $L$-functions in the weight $k$ aspect over $K\le k\le 2K$, with $K$ being a large number. As the results, we obtain a Lindelöf average bound for the first moment of $L(1/2, f)L(1/2, F\otimes f)$ of degree 8 and an asymptotic formula for the first moment of $L(1/2, F\otimes f)$ with an error term of $O(K^{-1/4+\varepsilon})$, respectively.

First moments of ${\rm{GL}} (3) \times {\rm{GL}} (2)$ and ${\rm{GL}} (2)$ $L$-functions and their applications

TL;DR

This work advances the subconvexity problem for high-riber L-functions by establishing first-moment bounds for GL(3)×GL(2) and GL(2) L-functions in both level and weight aspects. It combines amplification, Petersson trace theory, and GL(3) Voronoï summation with reciprocity to control intricate bilinear forms and Kloosterman sums, carefully addressing root-number contributions. The main contributions include a level-aspect subconvex bound for L(1/2, F\otimes f) in a hybrid prime-level range, a Lindelöf-mean bound for the weight- aspect first moment of L-factors, and a power-saving asymptotic formula in the weight variable K, with non-vanishing conclusions as a by-product. The results provide new tools for studying equidistribution of L^q-mass, Heegner points, and QUE for higher-rank automorphic L-functions, and they illustrate a path toward simultaneous subconvexity in multiple automorphic parameters.

Abstract

Let be a self-dual Hecke-Maaß form for underlying the symmetric square lift of a -newform of square-free level and trivial nebentypus. In this paper, we are interested in the first moments of the central values of -functions and -functions. As a result, we obtain an estimate for the first moment for over a family, where is of the level , and for any primes such that . We prove the subconvex bound for involving the levels aspects simultaneously in the range and for any for the first time. Moreover, we further investigate the first moments of these -functions in the weight aspect over , with being a large number. As the results, we obtain a Lindelöf average bound for the first moment of of degree 8 and an asymptotic formula for the first moment of with an error term of , respectively.
Paper Structure (19 sections, 17 theorems, 150 equations)

This paper contains 19 sections, 17 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Prerequisites
  4. Automorphic forms
  5. Automorphic $L$-functions and functional equations
  6. Epsilon constant
  7. Voronoı̆ summation formula
  8. Non-linear exponential sums
  9. Petersson trace formula
  10. Bounds on bilinear forms with Kloosterman sums
  11. Bessel functions
  12. Proofs of Theorem \ref{['3243251515']} and Corollary \ref{['corodoe923']}
  13. Proof of Theorem \ref{['3243251515']}
  14. Non-oscillatory case
  15. Oscillatory case
  16. ...and 4 more sections

Key Result

Theorem 1.2

Let $q,M\ge 2$ be two primes satisfying that $(q,M)=1$ and $M^{1/8}<q<M^{3/8}$. Let $F$ be a Hecke-Maaß form for the congruence subgroup $\Gamma_0(q^2)$ obtained as the symmetric square lift of a ${\mathrm{GL}}(2)$-newform of level $q$ with trivial nebentypus. Define the set with Then, for any $\ell =\{1,\ell_1\ell_2, \ell^2_1\ell^2_2\}$ with $\ell_1,\ell_2\in \mathscr{P}_L$, we have where the

Theorems & Definitions (21)

  • Conjecture 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 11 more